Parameter estimation and model reduction for model predictive control in retinal laser treatment. - In: Control engineering practice, ISSN 1873-6939, Bd. 128 (2022), 105320, S. 1-14
Laser photocoagulation is one of the most frequently used treatment approaches for retinal diseases such as diabetic retinopathy and macular edema. The use of model-based control, such as Model Predictive Control (MPC), enhances a safe and effective treatment by guaranteeing temperature bounds. In general, real-time requirements for model-based control designs are not met since the temperature distribution in the eye fundus is governed by a heat equation with a nonlinear parameter dependency. This issue is circumvented by representing the model by a lower-dimensional system which well-approximates the original model, including the parametric dependency. We combine a global-basis approach with the discrete empirical interpolation method, tailor its hyperparameters to laser photocoagulation, and show its superiority in comparison to a recently proposed method based on Taylor-series approximation. Its effectiveness is measured in computation time for MPC. We further present a case study to estimate the range of absorption parameters in porcine eyes, and by means of a theoretical and numerical sensitivity analysis we show that the sensitivity of the temperature increase is higher with respect to the absorption coefficient of the retinal pigment epithelium (RPE) than of the choroid’s.
https://doi.org/10.1016/j.conengprac.2022.105320
Eigenvalues of parametric rank one perturbations of matrix pencils. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2022. - 1 Online-Ressource (37 Seiten). - (Preprint ; M22,04)
The behavior of eigenvalues of regular matrix pencils under rank one perturbations which depend on a scalar parameter is studied. In particular we address the change of the algebraic multiplicities, the change of the eigenvalues for small parameter variations as well as the asymptotic eigenvalue behavior as the parameter tends to infinity. Besides that, an interlacing result for rank one perturbations of matrix pencils is obtained. Finally, we apply the result to a redesign problem for electrical circuits.
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2022200237
Model predictive control tailored to epidemic models. - In: 2022 European Control Conference (ECC), (2022), S. 743-748
We propose a model predictive control (MPC) approach for minimising the social distancing and quarantine measures during a pandemic while maintaining a hard infection cap. To this end, we study the admissible and the maximal robust positively invariant set (MRPI) of the standard SEIR compartmental model with control inputs. Exploiting the fact that in the MRPI all restrictions can be lifted without violating the infection cap, we choose a suitable subset of the MRPI to define terminal constraints in our MPC routine and show that the number of infected people decays exponentially within this set. Furthermore, under mild assumptions we prove existence of a uniform bound on the time required to reach this terminal region (without violating the infection cap) starting in the admissible set. The findings are substantiated based on a numerical case study.
https://doi.org/10.23919/ECC55457.2022.9838589
On sparse random combinatorial matrices. - In: Discrete mathematics, Bd. 345 (2022), 11, 113017
Let Qn,d denote the random combinatorial matrix whose rows are independent of one another and such that each row is sampled uniformly at random from the subset of vectors in {0,1}n having precisely d entries equal to 1. We present a short proof of the fact that P[det(Qn,d)=0]=O(n1/2log3/2nd)=o(1), whenever ω(n1/2log3/2n)=d≤n/2. In particular, our proof accommodates sparse random combinatorial matrices in the sense that d=o(n) is allowed. We also consider the singularity of deterministic integer matrices A randomly perturbed by a sparse combinatorial matrix. In particular, we prove that P[det(A+Qn,d)=0]=O(n1/2log3/2nd), again, whenever ω(n1/2log3/2n)=d≤n/2 and A has the property that (1,-d) is not an eigenpair of A.
https://doi.org/10.1016/j.disc.2022.113017
Reachability in arborescence packings. - In: Discrete applied mathematics, ISSN 1872-6771, Bd. 320 (2022), S. 170-183
Fortier et al. proposed several research problems on packing arborescences and settled some of them. Others were later solved by Matsuoka and Tanigawa and by Gao and Yang. The last open problem is settled in this article. We show how to turn an inductive idea used in the latter two articles into a simple proof technique that allows to relate previous results on arborescence packings. We prove that a strong version of Edmonds’ theorem on packing spanning arborescences implies Kamiyama, Katoh and Takizawa’s result on packing reachability arborescences and that Durand de Gevigney, Nguyen and Szigeti’s theorem on matroid-based packing of arborescences implies Király’s result on matroid-reachability-based packing of arborescences. Further, we deduce a new result on matroid-reachability-based packing of mixed hyperarborescences from a theorem on matroid-based packing of mixed hyperarborescences due to Fortier et al.. Finally, we deal with the algorithmic aspects of the problems considered. We first obtain algorithms to find the desired packings of arborescences in all settings and then apply Edmonds’ weighted matroid intersection algorithm to also find solutions minimizing a given weight function.
https://doi.org/10.1016/j.dam.2022.05.018
Good acyclic orientations of 4-regular 4-connected graphs. - In: Journal of graph theory, ISSN 1097-0118, Bd. 100 (2022), 4, S. 698-720
An st-ordering of a graph G=(V,E) is an ordering v1,v2,…,vn of its vertex set such that s=v1,t=vn and every vertex vi with i=2,3,…,n-1 has both a lower numbered and a higher numbered neighbor. Such orderings have played an important role in algorithms for planarity testing. It is well-known that every 2-connected graph has an st-ordering for every choice of distinct vertices s,t. An st-ordering of a graph G corresponds directly to a so-called bipolar orientation of G, that is, an acyclic orientation D of G in which s is the unique source and t is the unique sink. Clearly every bipolar orientation of a graph has an out-branching rooted at the source vertex and an in-branching rooted at the sink vertex. In this paper, we study graphs which admit a bipolar orientation that contains an out-branching and in-branching which are arc-disjoint (such an orientation is called good). A 2T-graph is a graph whose edge set can be decomposed into two edge-disjoint spanning trees. Clearly a graph has a good orientation if and only if it contains a spanning 2T-graph with a good orientation, implying that 2T-graphs play a central role. It is a well-known result due to Tutte and Nash-Williams, respectively, that every 4-edge-connected graph contains a spanning 2T-graph. Vertex-minimal 2T-graphs with at least two vertices, also known as generic circuits, play an important role in rigidity theory for graphs. Recently with Bessy and Huang we proved that every generic circuit has a good orientation. In fact, we may specify the roots of the two branchings arbitrarily as long as they are distinct. Using this, several results on good orientations of 2T-graphs were obtained. It is an open problem whether there exists a polynomial algorithm for deciding whether a given 2T-graph has a good orientation. Complex constructions of 2T-graphs with no good orientation were given in work by Bang-Jensen, Bessy, Huang and Kriesell (2021) indicating that the problem might be very difficult. In this paper, we focus on so-called quartics which are 2T-graphs where every vertex has degree 3 or 4. We identify a sufficient condition for a quartic to have a good orientation, give a polynomial algorithm to recognize quartics satisfying the condition and a polynomial algorithm to produce a good orientation when this condition is met. As a consequence of these results we prove that every 4-regular and 4-connected graph has a good orientation, where, as for generic circuits, we may specify the roots of the two branchings arbitrarily as long as they are distinct. We also provide evidence that even for quartics it may be difficult to find a characterization of those instances which have a good orientation. We also show that every graph on n≥8 vertices and of minimum degree at least has a good orientation. Finally we pose a number of open problems.
https://doi.org/10.1002/jgt.22803
Generalized boundary triples, II : some applications of generalized boundary triples and form domain invariant Nevanlinna functions. - In: Mathematische Nachrichten, ISSN 1522-2616, Bd. 295 (2022), 6, S. 1113-1162
The paper is a continuation of Part I and contains several further results on generalized boundary triples, the corresponding Weyl functions, and applications of this technique to ordinary and partial differential operators. We establish a connection between Post's theory of boundary pairs of closed nonnegative forms on the one hand and the theory of generalized boundary triples of nonnegative symmetric operators on the other hand. Applications to the Laplacian operator on bounded domains with smooth, Lipschitz, and even rough boundary, as well as to mixed boundary value problem for the Laplacian are given. Other applications concern with the momentum, Schrödinger, and Dirac operators with local point interactions. These operators demonstrate natural occurrence of ES$ES$-generalized boundary triples with domain invariant Weyl functions and essentially selfadjoint reference operators A0.
https://doi.org/10.1002/mana.202000049
Linear port-Hamiltonian systems are generically controllable. - In: IEEE transactions on automatic control, ISSN 1558-2523, Bd. 67 (2022), 6, S. 3220-3222
The new concept of relative generic subsets is introduced. It is shown that the set of controllable linear finite-dimensional port-Hamiltonian systems is a relative generic subset of the set of all linear finite-dimensional port-Hamiltonian systems. This implies that a random, continuously distributed port-Hamiltonian system is almost surely controllable.
https://doi.org/10.1109/TAC.2021.3098176
Funnel MPC with feasibility constraints for nonlinear systems with arbitrary relative degree. - In: IEEE control systems letters, ISSN 2475-1456, Bd. 6 (2022), S. 2804-2809
We study tracking control for nonlinear systems with known relative degree and stable internal dynamics by the recently introduced technique of Funnel MPC. The objective is to achieve the evolution of the tracking error within a prescribed performance funnel. We propose a novel stage cost for Funnel MPC, extending earlier designs to the case of arbitrary relative degree, and show that the control objective as well as initial and recursive feasibility are always achieved - without requiring any terminal conditions or a sufficiently long prediction horizon. We only impose an additional feasibility constraint in the optimal control problem.
https://doi.org/10.1109/LCSYS.2022.3178478
A vectorization scheme for nonconvex set optimization problems. - In: SIAM journal on optimization, ISSN 1095-7189, Bd. 32 (2022), 2, S. 1184-1209
In this paper, we study a solution approach for set optimization problems with respect to the lower set less relation. This approach can serve as a base for numerically solving set optimization problems by using established solvers from multiobjective optimization. Our strategy consists of deriving a parametric family of multiobjective optimization problems whose optimal solution sets approximate, in a specific sense, that of the set-valued problem with arbitrary accuracy. We also examine particular classes of set-valued mappings for which the corresponding set optimization problem is equivalent to a multiobjective optimization problem in the generated family. Surprisingly, this includes set-valued mappings with a convex graph.
https://doi.org/10.1137/21M143683X